Problem:
 a__zeros() -> cons(0(),zeros())
 a__and(tt(),X) -> mark(X)
 a__length(nil()) -> 0()
 a__length(cons(N,L)) -> s(a__length(mark(L)))
 a__take(0(),IL) -> nil()
 a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL))
 mark(zeros()) -> a__zeros()
 mark(and(X1,X2)) -> a__and(mark(X1),X2)
 mark(length(X)) -> a__length(mark(X))
 mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(0()) -> 0()
 mark(tt()) -> tt()
 mark(nil()) -> nil()
 mark(s(X)) -> s(mark(X))
 a__zeros() -> zeros()
 a__and(X1,X2) -> and(X1,X2)
 a__length(X) -> length(X)
 a__take(X1,X2) -> take(X1,X2)

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [length](x0) = x0,
   
   [and](x0, x1) = x0 + 4x1,
   
   [take](x0, x1) = x0 + x1,
   
   [a__take](x0, x1) = x0 + x1,
   
   [s](x0) = x0,
   
   [a__length](x0) = x0,
   
   [nil] = 0,
   
   [mark](x0) = x0,
   
   [a__and](x0, x1) = x0 + 4x1,
   
   [tt] = 1,
   
   [cons](x0, x1) = x0 + x1,
   
   [zeros] = 0,
   
   [0] = 0,
   
   [a__zeros] = 0
  orientation:
   a__zeros() = 0 >= 0 = cons(0(),zeros())
   
   a__and(tt(),X) = 4X + 1 >= X = mark(X)
   
   a__length(nil()) = 0 >= 0 = 0()
   
   a__length(cons(N,L)) = L + N >= L = s(a__length(mark(L)))
   
   a__take(0(),IL) = IL >= 0 = nil()
   
   a__take(s(M),cons(N,IL)) = IL + M + N >= IL + M + N = cons(mark(N),take(M,IL))
   
   mark(zeros()) = 0 >= 0 = a__zeros()
   
   mark(and(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__and(mark(X1),X2)
   
   mark(length(X)) = X >= X = a__length(mark(X))
   
   mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2))
   
   mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2)
   
   mark(0()) = 0 >= 0 = 0()
   
   mark(tt()) = 1 >= 1 = tt()
   
   mark(nil()) = 0 >= 0 = nil()
   
   mark(s(X)) = X >= X = s(mark(X))
   
   a__zeros() = 0 >= 0 = zeros()
   
   a__and(X1,X2) = X1 + 4X2 >= X1 + 4X2 = and(X1,X2)
   
   a__length(X) = X >= X = length(X)
   
   a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2)
  problem:
   a__zeros() -> cons(0(),zeros())
   a__length(nil()) -> 0()
   a__length(cons(N,L)) -> s(a__length(mark(L)))
   a__take(0(),IL) -> nil()
   a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL))
   mark(zeros()) -> a__zeros()
   mark(and(X1,X2)) -> a__and(mark(X1),X2)
   mark(length(X)) -> a__length(mark(X))
   mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   mark(0()) -> 0()
   mark(tt()) -> tt()
   mark(nil()) -> nil()
   mark(s(X)) -> s(mark(X))
   a__zeros() -> zeros()
   a__and(X1,X2) -> and(X1,X2)
   a__length(X) -> length(X)
   a__take(X1,X2) -> take(X1,X2)
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [length](x0) = 4x0 + 1,
    
    [and](x0, x1) = x0 + 2x1 + 2,
    
    [take](x0, x1) = x0 + x1,
    
    [a__take](x0, x1) = x0 + x1,
    
    [s](x0) = x0,
    
    [a__length](x0) = 4x0 + 1,
    
    [nil] = 0,
    
    [mark](x0) = x0,
    
    [a__and](x0, x1) = x0 + 2x1 + 2,
    
    [tt] = 3,
    
    [cons](x0, x1) = 2x0 + x1,
    
    [zeros] = 0,
    
    [0] = 0,
    
    [a__zeros] = 0
   orientation:
    a__zeros() = 0 >= 0 = cons(0(),zeros())
    
    a__length(nil()) = 1 >= 0 = 0()
    
    a__length(cons(N,L)) = 4L + 8N + 1 >= 4L + 1 = s(a__length(mark(L)))
    
    a__take(0(),IL) = IL >= 0 = nil()
    
    a__take(s(M),cons(N,IL)) = IL + M + 2N >= IL + M + 2N = cons(mark(N),take(M,IL))
    
    mark(zeros()) = 0 >= 0 = a__zeros()
    
    mark(and(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = a__and(mark(X1),X2)
    
    mark(length(X)) = 4X + 1 >= 4X + 1 = a__length(mark(X))
    
    mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2))
    
    mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2)
    
    mark(0()) = 0 >= 0 = 0()
    
    mark(tt()) = 3 >= 3 = tt()
    
    mark(nil()) = 0 >= 0 = nil()
    
    mark(s(X)) = X >= X = s(mark(X))
    
    a__zeros() = 0 >= 0 = zeros()
    
    a__and(X1,X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = and(X1,X2)
    
    a__length(X) = 4X + 1 >= 4X + 1 = length(X)
    
    a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2)
   problem:
    a__zeros() -> cons(0(),zeros())
    a__length(cons(N,L)) -> s(a__length(mark(L)))
    a__take(0(),IL) -> nil()
    a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL))
    mark(zeros()) -> a__zeros()
    mark(and(X1,X2)) -> a__and(mark(X1),X2)
    mark(length(X)) -> a__length(mark(X))
    mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    mark(0()) -> 0()
    mark(tt()) -> tt()
    mark(nil()) -> nil()
    mark(s(X)) -> s(mark(X))
    a__zeros() -> zeros()
    a__and(X1,X2) -> and(X1,X2)
    a__length(X) -> length(X)
    a__take(X1,X2) -> take(X1,X2)
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [length](x0) = x0,
     
     [and](x0, x1) = 4x0 + 4x1 + 1,
     
     [take](x0, x1) = x0 + x1 + 2,
     
     [a__take](x0, x1) = x0 + x1 + 2,
     
     [s](x0) = x0,
     
     [a__length](x0) = x0,
     
     [nil] = 1,
     
     [mark](x0) = x0,
     
     [a__and](x0, x1) = 4x0 + 4x1 + 1,
     
     [tt] = 1,
     
     [cons](x0, x1) = x0 + x1,
     
     [zeros] = 0,
     
     [0] = 0,
     
     [a__zeros] = 0
    orientation:
     a__zeros() = 0 >= 0 = cons(0(),zeros())
     
     a__length(cons(N,L)) = L + N >= L = s(a__length(mark(L)))
     
     a__take(0(),IL) = IL + 2 >= 1 = nil()
     
     a__take(s(M),cons(N,IL)) = IL + M + N + 2 >= IL + M + N + 2 = cons(mark(N),take(M,IL))
     
     mark(zeros()) = 0 >= 0 = a__zeros()
     
     mark(and(X1,X2)) = 4X1 + 4X2 + 1 >= 4X1 + 4X2 + 1 = a__and(mark(X1),X2)
     
     mark(length(X)) = X >= X = a__length(mark(X))
     
     mark(take(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = a__take(mark(X1),mark(X2))
     
     mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2)
     
     mark(0()) = 0 >= 0 = 0()
     
     mark(tt()) = 1 >= 1 = tt()
     
     mark(nil()) = 1 >= 1 = nil()
     
     mark(s(X)) = X >= X = s(mark(X))
     
     a__zeros() = 0 >= 0 = zeros()
     
     a__and(X1,X2) = 4X1 + 4X2 + 1 >= 4X1 + 4X2 + 1 = and(X1,X2)
     
     a__length(X) = X >= X = length(X)
     
     a__take(X1,X2) = X1 + X2 + 2 >= X1 + X2 + 2 = take(X1,X2)
    problem:
     a__zeros() -> cons(0(),zeros())
     a__length(cons(N,L)) -> s(a__length(mark(L)))
     a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL))
     mark(zeros()) -> a__zeros()
     mark(and(X1,X2)) -> a__and(mark(X1),X2)
     mark(length(X)) -> a__length(mark(X))
     mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
     mark(cons(X1,X2)) -> cons(mark(X1),X2)
     mark(0()) -> 0()
     mark(tt()) -> tt()
     mark(nil()) -> nil()
     mark(s(X)) -> s(mark(X))
     a__zeros() -> zeros()
     a__and(X1,X2) -> and(X1,X2)
     a__length(X) -> length(X)
     a__take(X1,X2) -> take(X1,X2)
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                     [1 1 0]     [0]
      [length](x0) = [0 1 0]x0 + [1]
                     [0 0 1]     [0],
      
                      [1 0 0]     [1 0 0]     [0]
      [and](x0, x1) = [0 0 0]x0 + [0 1 1]x1 + [0]
                      [0 0 1]     [0 0 0]     [1],
      
                            [1 1 0]  
      [take](x0, x1) = x0 + [0 1 0]x1
                            [0 0 1]  ,
      
                               [1 1 0]  
      [a__take](x0, x1) = x0 + [0 1 0]x1
                               [0 0 1]  ,
      
                [1 0 0]  
      [s](x0) = [0 1 1]x0
                [0 0 1]  ,
      
                        [1 1 0]     [0]
      [a__length](x0) = [0 1 0]x0 + [1]
                        [0 0 1]     [0],
      
              [0]
      [nil] = [0]
              [0],
      
                   [1 0 1]  
      [mark](x0) = [0 1 0]x0
                   [0 0 1]  ,
      
                         [1 0 0]     [1 0 0]     [0]
      [a__and](x0, x1) = [0 0 0]x0 + [0 1 1]x1 + [0]
                         [0 0 1]     [0 0 0]     [1],
      
             [0]
      [tt] = [0]
             [0],
      
                       [1 0 0]     [1 0 0]  
      [cons](x0, x1) = [0 0 1]x0 + [0 1 1]x1
                       [0 0 1]     [0 0 1]  ,
      
                [0]
      [zeros] = [0]
                [0],
      
            [0]
      [0] = [0]
            [0],
      
                   [0]
      [a__zeros] = [0]
                   [0]
     orientation:
                   [0]    [0]                    
      a__zeros() = [0] >= [0] = cons(0(),zeros())
                   [0]    [0]                    
      
                             [1 1 1]    [1 0 1]    [0]    [1 1 1]    [0]                        
      a__length(cons(N,L)) = [0 1 1]L + [0 0 1]N + [1] >= [0 1 1]L + [1] = s(a__length(mark(L)))
                             [0 0 1]    [0 0 1]    [0]    [0 0 1]    [0]                        
      
                                 [1 1 1]     [1 0 0]    [1 0 1]     [1 1 0]     [1 0 0]    [1 0 1]                            
      a__take(s(M),cons(N,IL)) = [0 1 1]IL + [0 1 1]M + [0 0 1]N >= [0 1 1]IL + [0 1 1]M + [0 0 1]N = cons(mark(N),take(M,IL))
                                 [0 0 1]     [0 0 1]    [0 0 1]     [0 0 1]     [0 0 1]    [0 0 1]                            
      
                      [0]    [0]             
      mark(zeros()) = [0] >= [0] = a__zeros()
                      [0]    [0]             
      
                         [1 0 1]     [1 0 0]     [1]    [1 0 1]     [1 0 0]     [0]                      
      mark(and(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 + [0] >= [0 0 0]X1 + [0 1 1]X2 + [0] = a__and(mark(X1),X2)
                         [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]                      
      
                        [1 1 1]    [0]    [1 1 1]    [0]                     
      mark(length(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__length(mark(X))
                        [0 0 1]    [0]    [0 0 1]    [0]                     
      
                          [1 0 1]     [1 1 1]      [1 0 1]     [1 1 1]                               
      mark(take(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__take(mark(X1),mark(X2))
                          [0 0 1]     [0 0 1]      [0 0 1]     [0 0 1]                               
      
                          [1 0 1]     [1 0 1]      [1 0 1]     [1 0 0]                      
      mark(cons(X1,X2)) = [0 0 1]X1 + [0 1 1]X2 >= [0 0 1]X1 + [0 1 1]X2 = cons(mark(X1),X2)
                          [0 0 1]     [0 0 1]      [0 0 1]     [0 0 1]                      
      
                  [0]    [0]      
      mark(0()) = [0] >= [0] = 0()
                  [0]    [0]      
      
                   [0]    [0]       
      mark(tt()) = [0] >= [0] = tt()
                   [0]    [0]       
      
                    [0]    [0]        
      mark(nil()) = [0] >= [0] = nil()
                    [0]    [0]        
      
                   [1 0 1]     [1 0 1]              
      mark(s(X)) = [0 1 1]X >= [0 1 1]X = s(mark(X))
                   [0 0 1]     [0 0 1]              
      
                   [0]    [0]          
      a__zeros() = [0] >= [0] = zeros()
                   [0]    [0]          
      
                      [1 0 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]             
      a__and(X1,X2) = [0 0 0]X1 + [0 1 1]X2 + [0] >= [0 0 0]X1 + [0 1 1]X2 + [0] = and(X1,X2)
                      [0 0 1]     [0 0 0]     [1]    [0 0 1]     [0 0 0]     [1]             
      
                     [1 1 0]    [0]    [1 1 0]    [0]            
      a__length(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = length(X)
                     [0 0 1]    [0]    [0 0 1]    [0]            
      
                            [1 1 0]           [1 1 0]                
      a__take(X1,X2) = X1 + [0 1 0]X2 >= X1 + [0 1 0]X2 = take(X1,X2)
                            [0 0 1]           [0 0 1]                
     problem:
      a__zeros() -> cons(0(),zeros())
      a__length(cons(N,L)) -> s(a__length(mark(L)))
      a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL))
      mark(zeros()) -> a__zeros()
      mark(length(X)) -> a__length(mark(X))
      mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
      mark(cons(X1,X2)) -> cons(mark(X1),X2)
      mark(0()) -> 0()
      mark(tt()) -> tt()
      mark(nil()) -> nil()
      mark(s(X)) -> s(mark(X))
      a__zeros() -> zeros()
      a__and(X1,X2) -> and(X1,X2)
      a__length(X) -> length(X)
      a__take(X1,X2) -> take(X1,X2)
     Unfolding Processor:
      loop length: 3
      terms:
       a__length(cons(N,zeros()))
       s(a__length(mark(zeros())))
       s(a__length(a__zeros()))
      context: s([])
      substitution:
       N -> 0()
      Qed